Abstract. We define a notion of Fourier-Mukai transform for abelian varieties on any oriented cohomology theory with Q-coefficients A∗Q . We use it to produce a Beauville decomposition of A∗Q and study its consequences, including a decomposition of the A-motive of an abelian variety.

1. Introduction The Fourier transformation is a well-known operator in analysis that gives an isometry between the L2 -spaces of a real vector space and its dual. In [12], Mukai introduced an analogous notion for ˆ be an abelian variety and its dual. Using sheaves of modules over abelian varieties. Let X and X ˆ the normalized Poincar´e bundle P on X × X Mukai defined a functor between the derived categories ˆ and proved that this is an equivalence of categories. In of the sheaves of modules over X and X, [2], Beauville used his results to define such a functor on cohomology, K-theory and the Chow ring of X with similar properties in each theory. He studied these Fourier transformations in detail and proved interesting consequences, among which is a decomposition of the Chow ring of an abelian variety into the eigenspaces of the pullbacks associated to the morphisms of multiplication by n, for all integers n (see [3]). Deninger and Murre used Beauville and Mukai’s work in [5] to give a decomposition of the diagonal in CH(X × X) ⊗ Q, which induces a canonical decomposition of Chow motives of an abelian variety. Levine and Morel introduced the notion of oriented cohomology theories on Smk (see [10] for the definition) and constructed the algebraic cobordism as the universal such theory. The Chow ring being an oriented cohomology theory, a natural question to ask is whether a functor with the usual properties of a Fourier-Mukai transformation can be defined in this more general setting. In this paper, we define a Fourier-Mukai transformation on any oriented cohomology theory with Q-coefficients and study its consequences. The key idea that helped us extend the definition of the Fourier-Mukai transformation is that, working with Q-coefficients, any oriented cohomology theory A∗ can be twisted to form another theory A∗ad which has an additive formal group law. Also, when applied on an abelian variety X, the triviality of the tangent bundle on X implies the existence of an isomorphism of rings A∗ (X) → A∗ad (X). Using these ideas, we generalize the Chern character and produce a ring ˆ After recalling homomorphism C : K 0 (X) → A∗Q (X). Let P denote the class of P in K 0 (X × X). some background material in sections 2 and 3, we use the “A-correspondence” C(P) in section 4 to ˆ so that it generalizes the known define the Fourier-Mukai transformation F A : A∗Q (X) → A∗Q (X), Fourier-Mukai transformations. In section 5, we obtain analogues of all the properties of F listed in [2, Proposition 3]. As a consequence, in section 6, we get a Beauville decomposition of A∗Q (X) for any oriented cohomology theory A∗ and any abelian variety X, generalizing [3, Th´eor`eme]. Theorem 1.1. Let X be an abelian variety of dimension g over k. Then, we have ApQ (X)

=

2p M

AQsp (X),

s=2p−2g

where

AQsp (X)

:= x ∈

ApQ (X)| ∀ n

∗

∈ Z, n x = n2p−s x .

2010 Mathematics Subject Classification. Primary 14F43; Secondary 55N22, 14K05, 14C25. Key words and phrases. algebraic cobordism, Fourier-Mukai transformation, Beauville decomposition, motivic decomposition. 1

2

ANANDAM BANERJEE AND THOMAS HUDSON

We also improve the limits of the decomposition in case of algebraic cobordism. In section 7, we prove that there is a canonical decomposition of the A-motive (defined in [14, § 6]) of an abelian variety. Theorem 1.2. There is a canonical decomposition of the A-motive with Q-coefficients of an abelian variety X of dimension g over k: 2g M hA (X) = hiA (X) , i=0 A where hA (X) = (X, idX , 0) is the motive of X, hiA (X) = (X, pA i , 0) and the pi ’s are such that A i A A cA (n) ◦ pi = n pi = pi ◦ cA (n).

In view of [8, Theorem B], we have a K¨ unneth isomorphism in MUQ . We check that the decomposition of the cobordism motive of an abelian variety is a K¨ unneth decomposition with respect to the canonical morphism from algebraic to complex cobordism. In section 8, we define an integral ˆ following the ideas in [2, Proposition 30 ]. As Fourier-Mukai transformation F A : A∗ (X) → A∗ (X) an application of the properties of the Fourier-Mukai transformation, in section 9 we generalize a result of Bloch [4] to the case of cobordism cycles. Let N ∗ (A) be the group of numerically trivial cobordism cycles on A as defined in [1, Definition 3.1] and let ? be the Pontrygin product on Ω∗ (A). Then, we show ∗?(g+1)

Proposition 1.3. NQ

(X) = (0).

As a corollary, we show that A∗?(g+1) (X) = 0 where A∗ (X) denotes the subgroup of Ω∗ (X) consisting of cobordism cycles algebraically equivalent to zero (see [7, § 3]). 2. Oriented cohomology theories and Algebraic cobordism In [10], inspired by the work of Quillen on complex differentiable manifolds, Levine and Morel introduced the notion of an oriented cohomology theory: a contravariant functor A∗ from Smk to graded rings together with a collection of push-forward maps f∗ associated to projective morphisms. This family is meant to respect functoriality and to be compatible on cartesian squares with the pull-back maps g ∗ every time two morphisms f and g are transverse. Finally, the functor is supposed to satisfy both the projective bundle formula, which expresses the evaluation of A∗ on a projective bundle in terms of that of the base, and the extended homotopy property, which requires p∗ : A∗ (X) → A∗ (V ) to be an isomorphism for every vector bundle E → V and every E-torsor p : V → X. As one might expect, a morphism of oriented cohomology theories is a natural transformation of functors which is also compatible with the push-forward morphisms f∗ . For the precise definition we refer the reader to [10, Definition 1.1.2]. Important examples of functors which are also oriented cohomology theories include the Chow ring CH ∗ and K 0 [β, β −1 ] := K 0 ⊗ Z[β, β −1 ], a graded version of the Grothendieck ring of vector bundles. A relevant feature of oriented cohomology theories is that they allow a theory of Chern classes. Even though in order to establish it for any bundle E → X it is necessary to rely on the projective bundle formula, the first Chern class of a line bundle L → X can be defined as c1 (L) := s∗ s∗ 1X ∈ A∗ (X), where s denotes the zero section of L. Once the first Chern class is available, one may consider how it relates to the tensor product. While for the Chow group one has c1 (L ⊗ M ) = c1 (L) + c1 (M ), the equality is not true in general for oriented cohomology theories and one is forced to replace the usual addition with a formal group law: c1 (L ⊗ M ) = FA (c1 (L), c1 (M )) ∗

for a certain FA ∈ A (k)[[u, v]]. A commutative formal group law of rank one (R, FR ) consists of a ring R and a power series FR ∈ R[[u, v]] satisfying conditions which are analogues of those for the operation in a group. For instance, the analogue of the associative property reads FR (FR (u, v), w) = FR (u, FR (v, w)) ∈ R[[u, v, w]] . In [9], Lazard identified the universal such object (L, F ) and proved that the ring of coefficients, now known as the Lazard ring, is isomorphic to Z[a1 , a2 , . . .]. In this context, the universality means

FOURIER-MUKAI TRANSFORMATION ON ALGEBRAIC COBORDISM

3

that for every formal group law (R, FR ) there exists a unique ring homomorphism φ(R,FR ) : L → R such that φ(R,FR ) (F ) = FR , where φ(R,FR ) (F ) stands for the power series obtained by applying φ(R,FR ) to the individual coefficients of F . Since it will be needed later on, let us add that the Lazard ring can be made into a graded ring L∗ by setting deg ai = −i. Taking into consideration formal group laws makes it evident that the analogy with the situation in topology does not end with the introduction of oriented cohomology theories. In fact, in [15], Quillen proved that complex cobordism M U ∗ is universal among complex oriented cohomology theories, that M U ∗ (pt) ' L∗ and finally that its formal group law is the universal one. From this perspective, the theory of algebraic cobordism Ω∗ , developed in [10] by Levine and Morel, represents the exact analogue of M U ∗ as testified by the following two theorems. Theorem 2.1 ([10, Theorem 1.2.6]). Let k be a field of characteristic 0. Then, given any oriented cohomology theory A∗ on Smk , there is a unique morphism νA : Ω∗ → A∗ of oriented cohomology theories. Theorem 2.2 ([10, Theorem 1.2.7]). For any field k of characteristic 0, the canonical homomorphism classifying FΩ φΩ : L∗ → Ω∗ (k) is an isomorphism. Notice that, provided Ω∗ (k) is identified with the Lazard ring via φΩ , the evaluation of νA on Spec k coincides with φ(A,FA ) and as a consequence one has νA (FΩ ) = FA . Before we briefly illustrate the construction of algebraic cobordism, let us recall the definition of ordinary and multiplicative theories, together with the associated analogues of Theorem 2.1. An oriented cohomology theory A∗ is said to be ordinary if FA = Fa is the additive formal group law, i.e. FA (u, v) = u + v. If instead there exists b ∈ A∗ (k), such that FA (u, v) = u + v − b · uv, then both A∗ and FA = Fm are said to be multiplicative. In case b happens to be a unit we will also say that A∗ is periodic. The following results describe the universal theories of these types, relating them to Ω∗ . Theorem 2.3 ([10, Theorem 1.2.2 & 7.1.4 (2)]). Let k be a field of characteristic 0. The theory Ω∗ ⊗L Z obtained by tensoring along φ(Z,Fa ) is isomorphic to CH ∗ . Moreover, if A∗ is an ordinary theory there exists a unique morphism of oriented cohomology theories CH νA : CH ∗ → A∗ .

Theorem 2.4 ([10, Theorem 1.2.3 & 7.1.4 (1)]). Let k be a field admitting resolution of singularities. The theory Ω∗ ⊗L Z[β, β −1 ] obtained by tensoring along φ(Z[β,β −1 ],Fm ) is isomorphic to K 0 [β, β −1 ]. Furthermore, if A∗ is a multiplicative periodic theory there exists a unique morphism of oriented cohomology theories K 0 [β,β −1 ]

νA

: K 0 [β, β −1 ] → A∗ .

Now we briefly sketch the construction of algebraic cobordism, following the presentation of [10, Chapter 2]. As a group Ω∗ (X) is obtained from the free group generated by the isomorphism classes of cobordism cycles [f : Y → X, L1 , . . . , Lr ] where f is a projective morphism with Y ∈ Smk and {L1 , . . . , Lr } is a (possibly empty) family of line bundles over Y . Such a cycle has dimension d = dimk Y − r and codimension dimk X − d. On this group one successively imposes three families of relations, each arising from a different geometric condition. For what concerns the multiplicative structure, it is achieved by constructing pull-backs for l.c.i. morphisms through an adaptation of the method used by Fulton for Chow groups: one relies on the deformation to the normal cone to reduce to the case of a divisor, which is handled separately.

4

ANANDAM BANERJEE AND THOMAS HUDSON

2.1. Twists of oriented cohomology theories. We recall from [10, § 4.1.8–9 & § 7.4.2] the construction of the twisting of an oriented cohomology theory on Q Smk . Let A∗ be such a theory P ∞ i+1 ∗ and define λ(τ ) (u) = i≥0 τi u ∈ A (k)[[u]], where τ = (τi ) ∈ i=0 A−i (k) and τ0 = 1. This −1 last condition ensures that λ(τ ) admits an inverse λ(τ ) . Definition 2.5. The inverse Todd class operator of a line bundle L → X is defined to be the operator on A∗ (X) given by the infinite sum −1

Tfdτ (L) =

∞ X

c˜1 (L)i τi .

i=0

In [10, Proposition 4.1.20], Levine and Morel showed that this definition can be extended uniquely to all vector bundles by imposing that for all exact sequences 0 → E 0 → E → E 00 → 0 over X one −1 −1 −1 has Tfd (E) = Tfd (E 0 ) ◦ Tfd (E 00 ). Thus, it naturally extends to a map τ

τ

τ

−1

Tfdτ : K 0 (X) → Aut(A∗ (X)). −1

In fact, on an oriented cohomology theory on Smk , Tfdτ (E) is simply the multiplication by the −1 element T d−1 (E) := Tfd (E)(1X ) ∈ A∗ (X), called the inverse Todd class of E. For any smooth τ

τ

f

−1 ∗ ∗ equidimensional Y → X, it is shown that T d−1 τ (f E) = f T dτ (E).

Suppose X, Y are in Smk . Then, any f : Y → X is an l.c.i. morphism. Let f = q ◦ i be a factorization such that i : Y → P is a regular embedding and q : P → X is smooth. Letting I be the ideal sheaf of Y in P , we define the normal bundle Ni to be the bundle over Y whose dual has I/I 2 as the sheaf of sections. Set Nf ∈ K 0 (Y ) to be the class [Ni ] − [i∗ Tq ], where Tq is the relative tangent bundle associated to q. For any τ as above, one may construct an oriented cohomology theory on Smk , denoted A∗(τ ) , by twisting the first Chern classes and the pull-back maps. If f ∗ and c1 are respectively the pull-backs and the first Chern class respectively in A∗ , then as groups A∗(τ ) (X) = A∗ (X) and in A∗(τ ) one has: −1 −1 ∗ ∗ • f(τ ) = T dτ (Nf ) · f , where T dτ is the inverse Todd class; (τ )

• for any line bundle L over X, the first Chern class of L in A∗(τ ) is c1 (L) = λ(τ ) (c1 (L)); • if · denotes the product in A∗ (X) and ·τ denotes the product in A∗(τ ) (X), then x ·τ y = ∗ T d−1 τ (NδX ) · x · y, for any x, y ∈ A (X), where δX : X → X × X is the diagonal morphism. Notice that the push-forward maps f∗ are unchanged and that the modification of the first Chern (τ ) −1 classes affects the formal group law which becomes FA (u, v) = λ(τ ) (FA (λ−1 (τ ) (u), λ(τ ) (v))). If f = q ◦ i is a factorization as before, then note that P is smooth over k, since X is smooth i / P , we get by [6, B.7.2.], the exact sequence and q is smooth. Thus, considering Y % y Spec k 0 −→ TY −→ i∗ TP −→ Ni −→ 0. Thus, in K 0 (Y ) one has [TY ] = [i∗ TP ] − [Ni ] and since q is smooth, [Tq ] = [TP ] − [q ∗ TX ]. Hence, Nf = [i∗ TP ] − [TY ] − i∗ ([TP ] − [q ∗ TX ]) = [f ∗ TX ] − [TY ]. Let us now consider more in detail the situation in the special case of abelian varieties: the tangent bundles TX and TY become trivial. Thus, f ∗ TX is trivial as well. It follows from the −1 properties of Tfd that τ

(2.1)

∗ −1 ∗ −1 ∗ −1 ∗ f(τ ) = T dτ (Nf ) · f = T dτ (f TX ) · T dτ (−TY ) · f −1 = 1A∗ (Y ) · (T d−1 · f ∗ = 1A∗ (Y ) · f ∗ = f ∗ . τ (TY ))

Note that, if X is an abelian variety, then T d−1 τ (NδX ) = 1A∗ (X) since X × X is also an abelian variety and δX is an l.c.i. morphism. Thus, we obtain the following: ∼

Lemma 2.6. For an abelian variety X, there is a ring isomorphism iτA : A∗(τ ) (X) → A∗ (X).

FOURIER-MUKAI TRANSFORMATION ON ALGEBRAIC COBORDISM

5

Proof. Since A∗ (X) and A∗(τ ) (X) coincide as groups, we only need to verify that the twisting does not affect the multiplicative structure. Let · denote the product in A∗ (X) and ·τ denote the product in A∗(τ ) (X). Then, for α, β ∈ A∗ (X), α ·τ β = T d−1 τ (NδX ) · α · β = α · β. Thus the map ∗ ∗ A(τ ) (X) → A (X) identifying the two groups is a ring isomorphism as well. Remark 2.7. Note that, even though iτA is an isomorphism, it is not true in general that iτA ([Y → X]A(τ ) ) = [Y → X]A . However, if π : X → Spec k is the structure morphism of an abelian variety, A

∗ ∗ A then the argument in (2.1) shows that iτA (1X(τ ) ) = iτA (π(τ ) (1k )) = π (1k ) = 1X since the tangent bundle on Spec k is also trivial.

Now we would like to highlight two applications of the twisting construction, which will be used in the definition of the generalized Fourier-Mukai transformation. As Levine and Morel point out in [10, Remark 4.2.11], the Chern character ch : K 0 → CHQ∗ can be recovered by making use of Theorem 2.4. By an appropriate twisting, CHQ∗ ⊗Z Z[β, β −1 ] can be made into a multiplicative theory and the theorem then provides a morphism of oriented cohomology theories whose degree 0 component is precisely ch. For our purposes we will need to consider a generalization of this morphism for all ordinary theories with rational coefficients. For any such theory A∗Q we define chA to be the composition CH chA := νAQ Q ◦ ch, CH

CH where νAQ Q is obtained from the morphism νA (arising from Theorem 2.3) by enlarging the CH

coefficient ring. Since νAQ Q is a morphism of oriented cohomology theories, it immediately follows that for all X ∈ Smk one has that chA (X) : K 0 (X) → A∗Q (X) is a ring homomorphism and that chA commutes with pushforward maps. Another application of the twisting construction is to associate to any theory with rational coefficients an ordinary one. In fact, [10, Lemma 4.1.29] shows that there exists η = (ηi ) ∈ Q∞ ∗ i=0 AQ (k) with η0 = 1, such that logA (u) = λ(η) is the logarithm of the formal group law FA , i.e. logA (FA (u, v)) = logA (u) + logA (v). The resulting theory A∗ad := A∗Q (η) is therefore ordinary and its first Chern classes are given by η ad ad the formula cA (L) = logA (cA 1 (L)). For simplicity the isomorphism iA will be denoted by iA . 1 It turns out to be useful to find the ηi ’s in terms of the coefficients of FA . X Lemma 2.8. If FA (u, v) = u + v + am,n (um v n + un v m ), then ηi ∈ A−i Q (k) and moreover m≥n≥1

(i + 1)ηi ∈ A−i (k). Proof. We proceed by comparing the coefficients of certain terms on both sides of the equality logA (FA (u, v)) = logA (u) + logA (v). By looking at the coefficient of uv, we get a1,1 + 2η1 = 0, a1,1 . Now consider the term uv i for some i ≥ 1. The only way for uv i to be a so that η1 = − 2 product of k terms is uv i−k+1 · v| ·{z · · v} for 1 ≤ k ≤ i + 1. Thus, the coefficient of uv i in the left k−1

i+1 X k hand side is ηk−1 ai−k+1,1 where we have a0,1 = 1. Equating this sum to 0, we have k−1 k=1 i X

(i + 1)ηi = −

kηk−1 ai−k+1,1 . Note that am,1 ∈ A−m (k). Thus, by applying induction, we may

k=1

conclude from the above that (i + 1)ηi ∈ A−i (k).

3. Recollection of A-motives In [14, § 5-6], for an oriented cohomology theory A∗ on Smk , Nenashev and Zainoulline constructed the A-motive of a smooth projective variety X over k, following the ideas of [11]. We briefly recall its construction.

6

ANANDAM BANERJEE AND THOMAS HUDSON

3.1. A-correspondences. Let X and Y be smooth projective varieties over an algebraically closed field k of characteristic 0. We recall from [14] some facts about the category of A-correspondences. Given an oriented cohomology theory A∗ , we define the category of A-correspondences, denoted CorA , as • Ob(CorA ) := Ob(SmProjk ); • HomCorA (X, Y ) := A∗ (X × Y ); • the composition of morphisms α ∈ A∗ (X × Y ) and β ∈ A∗ (Y × Z) is the correspondence β ◦ α := (pXZ )∗ (p∗XY (α) · p∗Y Z (β)) ∈ A∗ (X × Z). where pXZ , pXY and pY Z are the respective projections from X × Y × Z. There is a functor cA : SmProjop k → CorA given by cA (X) = X and cA (f ) = (Γf )∗ (1A(X) ) ∈ A∗ (Y × X) (f,id)

for a morphism f : X → Y , where Γf : X −→ Y × X is the graph morphism. For α ∈ A∗ (X × Y ), we have the transpose αt := ι∗ (α) ∈ A∗ (Y × X), where ι : Y × X → X × Y is given by swapping the variables. For a correspondence α ∈ HomCorA (Y, X), we define its realization RA (α) : A∗ (Y ) → A∗ (X) as follows: we identify A∗ (Y ) with HomCorA (pt, Y ) and note that α defines a map HomCorA (pt, Y ) → HomCorA (pt, X) given by composition with α. This defines the map RA (α) as β 7→ pX∗ (α · p∗Y β), where pX and pY are the respective projections to X and Y from Y × X. When no confusion is likely to arise we will replace RA by R. Note that the projection formula for the oriented cohomology theory A∗ implies that R(cA (f )) = f ∗ and R(cA (f )t ) = f∗ ,

(3.1)

Z so that the functor A∗ : SmProjop k → Ab factors through CorA . ∗ ∗ If α ∈ A (X × Y ) and β ∈ A (Y × Z), it follows from the definition that

(3.2)

R(β) ◦ R(α) = R(β ◦ α) = R (pXZ )∗ (p∗XY (α) · p∗Y Z (β)) ,

Also, using the Projection formula, one has (3.3)

cA (f ) ◦ α = (idZ × f )∗ (α) ∗

and

β ◦ cA (f ) = (f × idZ )∗ (β)

∗

for f : X → Y , α ∈ A (Z × Y ) and β ∈ A (X × Z). Applying transpose, we also get that for γ ∈ A∗ (Y × Z) and δ ∈ A∗ (Z × X), (3.4)

γ ◦ cA (f )t = (f × idZ )∗ (γ)

and

cA (f )t ◦ δ = (idZ × f )∗ (δ).

The grading on A∗ induces a grading on HomCorA which makes it into a graded algebra under composition. It is given as HomnCorA (X, Y ) := ⊕i An+di (Xi × Y ), where the Xi ’s are the irreducible components of X and ` di = dimXi . It is worth stressing that CorA forms an additive category by defining X ⊕ Y = X Y . 3.2. A-motives. 0 Definition 3.1. Consider the category CorA with the same objects as CorA and HomCorA0 (X, Y ) := 0 0 HomCorA (X, Y ). The pseudo-abelian completion of CorA is called the category of effective Aeff motives, denoted by MA . This means that the objects in Meff A are pairs (X, p) where X ∈ Ob(CorA ) and p ∈ HomCorA0 (X, X) is a projector (that is, p ◦ p = p) and that the morphisms are given by {α ∈ HomCorA0 (X, Y )|α ◦ p = q ◦ α} . HomMeff ((X, p), (Y, q)) = A {α ∈ HomCorA0 (X, Y )|α ◦ p = q ◦ α = 0}

FOURIER-MUKAI TRANSFORMATION ON ALGEBRAIC COBORDISM

7

The category of A-motives, denoted by MA , has as objects triplets (X, p, m) where (X, p) is an object in Meff A and m ∈ Z. The morphisms are defined as: HomMA ((X, p, m), (Y, q, n)) =

{α ∈ Homn−m CorA (X, Y )|α ◦ p = q ◦ α} {α ∈ Homn−m CorA (X, Y )|α ◦ p = q ◦ α = 0}

.

Note that this means id(X,p,0) = idX = p ∈ HomMA ((X, p, 0), (X, p, 0)). We abuse notation to write idX to mean cA (idX ) ∈ Hom0CorA (X, X). The motive (X, idX , 0) is called the motive of X and is denoted by hA (X). The additive structure of CorA induces a direct sum in the category MA . 4. Fourier-Mukai operator π

X Spec k 4.1. Notation for Abelian varieties. From now on, unless stated otherwise, X −→ ˆ will be an abelian variety over k of dimension g. Its dual abelian variety will be denoted X ˆ will represent the normalized Poincar´e bundle with P being its class in the and P → X × X Grothendieck ring of vector bundles. Here, “normalized” means that i∗ P and ˆi∗ P are trivial, ˆ → X×X ˆ and ˆi : X × {ˆ0} → X × X ˆ are inclusions. By definition X comes where i : {0} × X equipped with a group operation µ : X × X → X, whose associated inverse morphism is denoted by σX . For any integer m we will write m : X → X to represent the morphism of multiplication by m with respect to the operation µ. As a general principle, we will add a ˆ to denote the corresponding notion for the dual abelian variety. Finally, let us recall that for any oriented cohomology theory A∗ it is possible to define on ∗ A (X) the so-called Pontryagin product. It will be denoted by ? and it is defined as

x ? y := µ∗ (p∗1 x · p∗2 y), where p1 and p2 are the projections of X × X onto the first and second factor, and · is the usual product on A∗ . 4.2. The definition of F. For any oriented cohomology theory A∗ we wish to define an operator ˆ which has the usual properties of the Fourier-Mukai transformation on F A : A∗Q (X) → A∗Q (X) Chow rings or K-theory (see [2]). The key observation that allows one to extend the definition given by Beauville is that the Fourier-Mukai transformation for CHQ∗ can be restated in terms of correspondences: F CH = R(ch(P)). Hence one is left with the task of indentifying the correct ˆ Our proposal is to make use of the twisting construction to reduce analogue of ch(P) in A∗ (X × X). to the case of an ordinary theory and then use the universality of CH ∗ among such theories. This leads to the definition of CA : K 0 (X) → A∗Q (X) as CA := iad AQ ◦ chA . It follows directly from the definition that for any line bundle L on X we have (4.1)

CA ([L]) = exp(logA (cA 1 (L))).

Proposition 4.1. Let A∗ be any oriented cohomology theory. Then, (1) CA is a ring homomorphism; (2) CA commutes with pullbacks of morphisms between abelian varieties; (3) CA commutes with pushforwards of morphisms between abelian varieties. Proof. The first statement is obvious since CA is the composition of two rings homomorphisms. CHQ For the second and the third stament one first recalls that by definition CA = iad AQ ◦ νAQ ◦ ch, so it suffices to prove that the statements hold for each of the morphisms separately. (2) holds for iad AQ in view of (2.1), while (3) follows since the twisting does not modify the pushforwards. On the other CH hand, νA is a morphism of oriented cohomology theories and hence by definition commutes with Q both pushforwards and pullbacks. Finally both statements are verified for the Chern character as well, since we are dealing with abelian varieties, see [6, §15.1 and Theorem 15.2]. Definition 4.2. Let A∗ be any oriented cohomology theory. We define the Fourier-Mukai opˆ to be F A := R(CA (P)). The dual operator Fˆ A is defined to be erator F A : A∗Q (X) → A∗Q (X) t R((CA (P)) ). When there is no confusion, we will denote F A and Fˆ A by F and Fˆ respectively.

8

ANANDAM BANERJEE AND THOMAS HUDSON

The operators F and Fˆ defined above satisfy the following: ∗ Fˆ ◦ F = (−1)g σX ∗ F ◦ Fˆ = (−1)g σX ˆ, : X → X is the multiplication by (−1) in the abelian variety X.

Theorem 4.3. We have where σX

∗ ∗ Proof. It follows from (3.2) that Fˆ ◦ F = R q13∗ (q12 (CA (P)) · q23 (ι∗ CA (P))) where qij are the ˆ × X. We may rewrite this as projections from X × X Fˆ ◦ F = R p12∗ (p∗13 (CA (P)) · p∗23 (CA (P))) ˆ It follows from Proposition 4.1 that where pij are the respective projections from X × X × X. ∗ ∗ ∗ ∗ p12∗ (p13 (CA (P)) · p23 (CA (P))) = CA (p12∗ [p13 P ⊗ p23 P]). But, by the Theorem of the Cube ([13, ˆ § 6]), p∗13 P⊗p∗23 P ' (µ×idXˆ )∗ P. Indeed, by the Theorem, it is enough to verify this on {0}×X×X, ˆ and X × X × {ˆ X × {0} × X 0}, where it follows from the properties of the Poincar´e bundle. Also, ˆ → X are transverse morphisms, we have p12∗ (µ × id ˆ )∗ P = µ∗ p1∗ P. But, since µ and p1 : X × X X ˆ p1∗ P = P (−1)i [Ri p1∗ P]. From [13, § 13] we have that Ri p1∗ P = 0 for i 6= g in K 0 (X × X), i ∼ and Rg p1∗ P = k(0) where k(0) is the sky-scrapper sheaf at the point 0 ∈ X. Note that µ∗ k(0) → ∼ 0 K OΓσX (X) in K (X). Since ΓσX is a closed immersion, OΓσX (X) → ΓσX ∗ 1X . Putting all these g K together, CA (µ∗ p1∗ P) = CA ((−1)g ΓσX ∗ 1K X ) = (−1) ΓσX ∗ CA (1X ). Finally, in view of Remark 2.7 g ∗ A K . The we have CA (1X ) = 1X , therefore Fˆ ◦ F = R((−1) c(σX )) = (−1)g R(c(σX )) = (−1)g σX other part is proved analogously.

5. Properties of the Fourier-Mukai operator We will need the following lemma: Lemma 5.1. Let f : X → Y be a finite surjective morphism between abelian varieties X and Y . Then, for any x ∈ A∗Q (X), f∗ f ∗ x = (deg f ) · x

∈ A∗Q (X),

where deg is the degree of a morphism. A Proof. By the projection formula, we get f∗ f ∗ x = f∗ (f ∗ x · 1A X ) = x · f∗ (1X ). Note that by Aad Aad CH CH A CH CH ad ad Remark 2.7, iA (f∗ 1X ) = f∗ 1X . But, f∗ 1X = νAad (f∗ 1X ) = νAad ((deg f )·1X ) = (deg f )·1A X , which completes the proof.

The Fourier-Mukai operator on A∗Q of abelian varieties satisfies the following properties: Proposition 5.2.

(1) For x, y ∈ A∗Q (X), we have

F A (x ? y) = F A (x)F A (y)

and

F A (xy) = (−1)g F A (x) ? F A (y).

ˆ be the dual isogeny. (2) Let f : X → Y be an isogeny of abelian varieties, and fˆ : Yˆ → X Then one has A A FYA ◦ f∗ = fˆ∗ ◦ FX and FX ◦ f ∗ = fˆ∗ ◦ FYA . X ˆ Then, for n ∈ Z, one has (3) Let x ∈ ApQ (X). Write F A (x) = yq , where yq ∈ AqQ (X). q

n∗ yq = ng−p+q yq , ˆ where n denotes the multiplication by n on X.

FOURIER-MUKAI TRANSFORMATION ON ALGEBRAIC COBORDISM

9

Proof. (1): First, we want to prove F A (x ? y) = F A (x)F A (y). Note that we have the following commutative diagram. ˆ X ×X ×X

µ×id

ˆ / X ×X

p12

X ×X

pX ˆ

ˆ /X

pX

/X

µ p2

p1

X '

X

ˆ Also, denote by πi , the projection Let p12 , p23 and p13 be the respective projections of X×X×X. ˆ to the i-th factor, for i = 1, 2, 3. By definition we have that x ? y = µ∗ (p∗ x · p∗ y), of X × X × X 1 2 so we obtain the following sequence of equalities. ∗ ∗ ∗ ∗ ∗ ∗ F A (x ? y) = pX∗ ˆ CA (P) · pX µ∗ (p1 x · p2 y) = pX∗ ˆ CA (P · (µ × idX )∗ p12 (p1 x · p2 y)) = π3∗ p∗13 CA (P) · p∗23 CA (P) · π1∗ x · π2∗ y [Since (µ × idX )∗ P = p∗13 P ⊗ p∗23 P] ∗ ∗ ∗ ∗ = π3∗ p∗13 (CA (P) · p∗X x) · p∗23 (CA (P) · p∗X y) = pX∗ ˆ p13∗ p13 (CA (P) · pX x) · p23 (CA (P) · pX y) ∗ ∗ ∗ ∗ ∗ ∗ = pX∗ ˆ (CA (P) · pX y) ˆ (CA (P) · pX x) · p13∗ p23 (CA (P) · pX y) = pX∗ ˆ (CA (P) · pX x) · pX ˆ pX∗ This is true since p13 and p23 are transverse morphisms. Thus, applying the Projection formula yields ∗ ∗ A A F A (x ? y) = pX∗ ˆ CA (P) · pX x · pX∗ ˆ CA (P) · pX y = R(CA (P)(x)) · R(CA (P)(y)) = F (x) · F (y). ˆ we have Fˆ A (x0 ? y 0 ) = Fˆ A (x0 ) · Fˆ A (y 0 ). Note Similarly, we can also show that for x0 , y 0 ∈ A∗Q (X), that, in view of Theorem 4.3, this implies the other statement. Indeed, ∗ A A g A A g A ˆA ˆA A σX F (F A (x)) · Fˆ A (F A (y)) ˆ (F (x) ? F (y)) = (−1) F ◦ F (F (x) ? F (y)) = (−1) F A ∗ ∗ = (−1)g F A σX (xy) = F A Fˆ A ◦ F A (xy) = (−1)g σX ˆ F (xy) Since σXˆ ◦ σXˆ = idXˆ , the result follows. (2): The combined use of (3.1), (3.2), (3.4) and (3.3) shows that FYA ◦ f∗ =R(CA (PY )) ◦ R(c(f )t ) = R(CA (PY ) ◦ c(f )t ) = R((f × idYˆ )∗ CA (PY )) and A fˆ∗ ◦ FX =R(c(fˆ)) ◦ R(CA (PX )) = R(c(fˆ) ◦ CA (PX )) = R((idX × fˆ)∗ CA (PX )).

But, by definition of fˆ, (idX × fˆ)∗ PX = (f × idYˆ )∗ PY . Similarly, the second assertion will follow if we show that (f × idXˆ )∗ CA (PX ) = (idY × fˆ)∗ CA (PY ). To see this, note that the transversality of the square X × Yˆ

idX ×fˆ

ˆ / X ×X

f ×idYˆ

Y × Yˆ

f ×idX ˆ

ˆ / Y ×X

idY ×fˆ

gives us in A∗Q (X) that (idY × fˆ)∗ (f × idXˆ )∗ CA (PX ) = (f × idYˆ )∗ (idX × fˆ)∗ CA (PX ) = (f × idYˆ )∗ (f × idYˆ )∗ CA (PY ) = (deg f )CA (PY )

[By Lemma 5.1, since deg(f × idYˆ ) = deg f ].

To finish the proof it suffices to apply (idY × fˆ)∗ to both sides and use Lemma 5.1. ˆ We first note that the Theorem of the (3): Consider the endomorphism (1X , n) of X × X. Squares ([13, II.6 - Corollary 4]) implies that (1X , n)∗ P = P⊗n . Indeed, as the bundle is rigidified along zero sections, the theorem gives us (1X , n)∗ P = (1X , (n − 1))∗ P ⊗ (1X , (n − 1))∗ P ⊗ (1X , (n − 2))∗ P−1

10

ANANDAM BANERJEE AND THOMAS HUDSON

and we have the above by induction. Aad (P)), one gets Since CA (P) = exp(logA (cA 1 (P))) = exp(c1 ad ad (1X , n)∗ CA (P) = exp(cA (P⊗n )) = exp(ncA (P)) = 1 1

2g X ni i=0

i!

ad (cA (P))i . 1

Thus, we have ∗ ∗ ∗ ∗ n∗ F A (x) = n∗ pX∗ ˆ pX x · CA (P) = pX∗ ˆ (1X , n) pX x · (1X , n) CA (P) ∗ = pX∗ ˆ pX x ·

2g X ni (cAad (P))i 1

i!

i=0 ∗ Note that pX∗ ˆ pX x ·

ad (P))i (cA 1

i!

=

2g X

∗ ni pX∗ ˆ pX x ·

i=0

ad (cA (P))i 1 i!

= yp+i−g . It now suffices to set q = p + i − g to obtain n∗ F A (x) =

p+g X

ng−p+q yq ,

q=p−g

which gives the desired equality.

6. Beauville decomposition for oriented cohomology theories We follow the ideas in [3] to give a decomposition of A∗Q (X) into eigenspaces of n∗ using the Fourier-Mukai operator defined in § 4. For s ∈ Z, let AQsp (X) denote the sub-group AQsp (X) := x ∈ ApQ (X)| ∀ n ∈ Z, n∗ x = n2p−s x . Proposition 6.1. Let x ∈ ApQ (X), and m be any integer other than 0, 1 or −1. The following conditions are equivalent: ˆ (1) F A (x) ∈ Ag−p+s (X); (2) x ∈ AQsp (X); Q (3) m∗ x = m2p−s x; (4) m∗ x = m2g−2p+s x; g−p+s ˆ A (5) F (x) ∈ AQs (X). X Proof. (1)⇒(2): Let y = (−1)g (σXˆ )∗ F A (x) and let Fˆ A (y) = xq with xq ∈ AqQ (X). q

Then, Proposition 5.2, part (3) gives us n∗ xq = ng−(g−p+s)+q xq = np+q−s xq . But, by Proposition 5.2, part (2) and Theorem 4.3, we get Fˆ A (y) = (−1)g Fˆ A ◦ (σXˆ )∗ ◦ F A (x) = (−1)g (σX )∗ ◦ Fˆ A ◦ F A (x) = (σX )∗ (σX )∗ (x) = x,

[By Lemma 5.1, since deg(σX ) = 1.]

Then, x = xp and n∗ x = n2p−s x, thus showing that x ∈ AQsp (X). (2)⇒(3): This is by definition. (3)⇒(4): Since deg m = m2g , in view of Lemma 5.1 one has m∗ m∗ x = m2g x. (4)⇒(5): By Proposition 5.2, part (2), we get that m∗ F A (x) = F A (m∗ x) = m2g−2p+s F A (x) = mg−p+(g−p+s) F A (x). ˆ Since m 6= 0, ±1, this implies by Proposition 5.2, part (3), that F A (x) ∈ Ag−p+s (X). Q g−p+s ˆ A (X). Then, by definition, F (x) ∈ AQs (5)⇒(1): This is obvious. Theorem 6.2. Let X be an abelian variety of dimension g over k. Then, we have ApQ (X) =

2p M s=2p−2g

AQsp (X).

FOURIER-MUKAI TRANSFORMATION ON ALGEBRAIC COBORDISM

11

P q ˆ Proof. Let x ∈ ApQ (X) and let y = F A (x). We can write y = q yq , with yq ∈ AQ (X). By q ˆ Then, Proposition 6.1 Proposition 5.2, part (3), n∗ yq = ng−p+q yq and hence yq ∈ AQp+q−g (X). X p gives us that Fˆ A (yq ) ∈ A (X). But, (−1)g (σX )∗ x = Fˆ A (y) = Fˆ A (yq ). Qp+q−g

q

! Aad i (c (P)) Since Fˆ A (yq ) has degree p and by definition Fˆ A (yq ) = p1∗ p∗2 (yq ) 1 , we get i! i≥0 ! ! ad ad (cA (P))p+g−q (cA (P))q+g−p A ∗ ∗ 1 1 ˆ F (yq ) = p1∗ p2 (yq ) . Also, yq = p2∗ p1 (x) since yq has de(p + g − q)! (q + g − p)! X

ad gree q. Now we notice that (cA (P))i vanishes unless 0 ≤ i ≤ 2g and this implies that yq and 1 A ˆ F (yq ) are both nonzero only if p − g ≤ q ≤ p + g. Since s + g − p = q, one immediately obtains the bounds, i.e. 2p − 2g ≤ s ≤ 2p.

Remark 6.3. For Ω∗Q the previous bound can be improved by observing that, since dimX = g, for P all x ∈ Ωp (X) one has F Ω (x) = q≤g yq . This allows one to conclude that 2p−g ≤ s ≤ Min(2p, p). The same observation holds for all theories which can be obtained from algebraic cobordism by imposing a formal group law, provided that the coefficient ring of definition has no elements of positive degree. For instance, this is the case for the Chow ring. As an easy consequence of the definition of AQsp (X) and Proposition 6.1, we get the following: ˆ Proposition 6.4. (1) F A (AQsp (X)) = AQsg−p+s (X). (2) If f : X → Y is a homomorphism of abelian varieties of relative dimension m, then f ∗ AQsp (Y ) ⊂ AQsp (X) and f∗ AQsp (X) ⊂ AQsp+m (Y ). p+q p+q−g (3) If x ∈ AQsp (X), y ∈ AQtq (X), then xy ∈ AQs+t (X) and x ? y ∈ AQs+t (X). Proof. (1) is immediate from Proposition 6.1. (2) follows from the fact that f ◦m = m◦f and the equivalence of (3) and (4) in Proposition 6.1. p+q Finally, if x ∈ AQsp (X) and y ∈ AQtq (X), then by definition xy ∈ AQs+t (X). Also, note that 2g−p−q+s+t ˆ p+q−g A A A A F (x ? y) = F (x)F (y) ∈ AQs+t (X). Applying Fˆ , we get (σX )∗ (x ? y) ∈ AQs+t (X), which gives the result by part (2).

6.1. Application to K-theory. Note that the Fourier-Mukai transformation that Beauville defined on K-theory in [2, § 1] does not lead to a decomposition of K 0 as there is no grading. However, Beauville’s work on the Chow ring induces one via the usual Chern character ch. We now want to explicitly express this decomposition by using Theorem 6.2 for the oriented cohomology theory K 0 [β, β −1 ]Q and consider its zero-component. Applying the argument in the proof of Lemma 2.8 to A∗ = K 0 [β, β −1 ]∗ readily gives us ηi = ∞ X β i−1 i βi , so that logK 0 [β,β −1 ] (x) = x for x ∈ K 0 [β, β −1 ]∗ (X). Thus, i+1 i i=1

K 0 [β,β −1 ]ad

c1

(P) = logK 0 [β,β −1 ] (β −1 (1 − [P∨ ])) = β −1

∞ X (1 − [P∨ ])i i=1

where log is the usual logarithm series.

i

= β −1 log([P∨ ])

12

ANANDAM BANERJEE AND THOMAS HUDSON

P 0 −1 Now, pick any x = [E]β −p ∈ K 0 [β, β −1 ]pQ (X). If F K [β,β ] (x) = q yq , we have from the proof of Theorem 6.2 that −1 log([P∨ ]))q+g−p β −q ∗ −p (β yq = p2∗ p1 ([E]β ) = p2∗ [p∗1 E](log([P∨ ]))q+g−p , (q + g − p)! (q + g − p)! 0 −1 (β −1 log([P∨ ]))p+g−q and Fˆ K [β,β ] (yq ) = p1∗ p∗2 (yq ) (p + g − q)! β −q (log([P∨ ]))p+g−q q−g−p = p1∗ p∗2 p2∗ ([p∗1 E](log([P∨ ]))q+g−p ) β (q + g − p)! (p + g − q)! β −p = p1∗ p∗2 p2∗ ([p∗1 E](log([P∨ ]))q+g−p )(log([P∨ ]))p+g−q . (q + g − p)!(p + g − q)!

7. Motivic decomposition Our goal in this section is to get a canonical decomposition of A-motives of abelian varieties as the one constructed for Chow motives byM Deninger and Murre in [5]. For an abelian variety X, we want to have a decomposition hA (X) = hiA (X) where hiA (X) = (X, pi , 0), pi being orthogonal i

projectors such that n∗ pi = ni pi . In [16, § 5], Scholl gave an alternative proof of the decomposition for Chow motives and also described the projectors in the decomposition more explicitly. Let ∆A denote the class of the diagonal morphism [X → X × X]A = cA (idX ) in AgQ (X × X). We are going to show Theorem 7.1. There is a canonical decomposition ∆A =

2g X

g pA i in AQ (X × X)

i=0 A A such that the pA i ’s are mutually orthogonal projectors ( i.e., pi ◦ pj = 0 for i 6= i A A i A A (idX × n)∗ pA i = n pi for all n ∈ Z and for which cA (n) ◦ pi = n pi = pi ◦ cA (n).

j) satisfying

Proof. Note that such a decomposition is unique if it exists. Indeed, if {qi }2g i=0 is another such P2g A decomposition, then pA = p ◦ q . It suffices to compose with c (n) from the left to get j A i j=0 i 2g X P2g j A A ni p A (nj − ni )pA i ◦ qj = 0. i = j=0 n pi ◦ qj , which by substituting the expression for pi gives j=0 A A Since, this is true for all n, we must have pA i ◦ qj = 0 for i 6= j, implying pi = pi ◦ qi . We can A similarly show that qi = pi ◦ qi and therefore the two decompositions coincide. To see the existence, we consider the following two cases: Case 1: A is an ordinary theory. As shown in [16, § 5], in CHQg (X × X), the diagonal ∆CH P2g may be expressed as ∆CH = i=0 pcan where for each i, pcan ◦ pcan = pcan and pcan ◦ pcan = 0 in i i i i i j can i can can HomCor0 ∗ (X, X) for i 6= j. Also, cCH (n) ◦ pi = n pi = pi ◦ cCH (n). CH Q

CH can CH Now, set pA is a morphism of oriented cohomology theories. i := νA (pi ). Note that νA ∗ CH CH CH Thus, for α, β ∈ CHQ (X × X), we have νA (α ◦ β) = νA (α) ◦ νA (β), so it readily follows that A A A CH pi is a projector and pi ◦ pj = 0 for i 6= j. Moreover, since, νA (cA (n)) = cCH (n), we also get i A A cA (n) ◦ pA i = n pi = pi ◦ cA (n) for all n ∈ Z. ∼ ∗ Case 2: A is not ordinary. In this case, since X is an abelian variety, iad AQ : Aad (X × X) → A∗Q (X × X) is a ring isomorphism, that commutes with pushforwards and pullbacks of morphisms ad between abelian varieties. This implies that for α, β ∈ A∗ad (X × X), we have iad AQ (α ◦ β) = iAQ (α) ◦ Aad A ad iad ) for all i gives us the desired result. AQ (β). Then, taking pi = iAQ (pi

Let MAQ be the category of A-motives over k with Q-coefficients, defined in section 3.2.

FOURIER-MUKAI TRANSFORMATION ON ALGEBRAIC COBORDISM

13

Corollary 7.2. In the category MAQ , there is a canonical decomposition: hA (X) =

2g M

hiA (X) ,

i=0 A where hA (X) = (X, idX , 0) is the motive of X, hiA (X) = (X, pA i , 0) and the pi ’s are such that A i A A cA (n) ◦ pi = n pi = pi ◦ cA (n).

Proof. This is immediate since, if cA (idX ) = Ln (X, idX , 0) = i=0 (X, pA i , 0).

n X

A pA i for mutually orthogonal projectors pi , then

i=0

Let νA : Ω∗ → A∗ denote the canonical morphism. νA induces a functor νf A : MΩ → MA of the corresponding categories of motives, acting as νA on the morphisms and on objects as (X, p, m) 7→ (X, νA (p), m). i i Corollary 7.3. We have νf A (hΩ (X)) = hA (X). A Proof. It follows from the construction in the proof of Theorem 7.1 that νA (pΩ i ) = pi for all i, can i i . which implies νf A (hΩ (X)) = hA (X) t A can Remark 7.4. We have (pA i ) = p2g−i since this property holds for the pi s.

Let MU∗ be complex cobordism theory. It follows from [8, Theorem B] that in MU∗Q , we have α the K¨ unneth isomorphism MU∗Q (Y ) ⊗L∗ MU∗Q (Z) → MU∗Q (Y × Z) for smooth projective varieties Y and Z over C. X Proposition 7.5. Let X be an abelian variety of dimension g over C. If α−1 (∆MU ) = aj ⊗ bi i+j=2g

with aj , bj ∈ MUjQ (X), then pMU = α(a2g−i ⊗ bi ). i Proof. For convenience, let us denote pi := pMU for each 0 ≤ i ≤ 2g. By Theorem 7.1, i

2g X

pi =

i=0 2g X

α(a2g−i ⊗ bi ) =

i=0

2g X

a2g−i × bi . Applying (id × n)∗ , we get

i=0 2g X

ni p i =

i=0

2g X i=0

a2g−i × n∗ bi =

2g X i=0

ni

X

a2g−j × bij

06j6i

P2g where, by Theorem 6.2, we have bi = j=i bji such that n∗ bji = nj bji . Since the above equality holds Pi P2g−i for all n, we obtain that pi = j=0 a2g−j × bij . This implies that pt2g−i = j=0 b2g−i × a2g−j = j P2g 2g−i t j=i b2g−j × aj . Thus, pi = p2g−i =⇒ 2g−i 2g−i a2g × bi0 + a2g−1 × bi1 + · · · + a2g−i × bii − b2g−i × a2g = 0. 2g−i × ai − b2g−i−1 × ai+1 − · · · − b0

=⇒a2g−j × bij = 0, for i > j, and Then, a2g−i × bi =

P2g

j=i

2g−i b2g−j × aj = 0, for i < j.

a2g−i × bji = a2g−i × bii = pi , which finishes the proof.

Remark 7.6. The decomposition of the cobordism motive of an abelian variety X is a cobordismK¨ unneth decomposition, in the sense that νMU (pΩ unneth component of i ) is the corresponding K¨ the diagonal in MU∗Q (X × X).

14

ANANDAM BANERJEE AND THOMAS HUDSON

8. Integral Fourier-Mukai transformation In this section, we define a Fourier-Mukai transformation on an oriented cohomology theory A∗ integrally, without Q-coefficients. We follow the ideas in [2, Proposition 30 ]. The key observation is that, for any x ∈ A∗ (X), F A (x) may be multiplied with a large enough integer N such that ˆ N F A (x) ∈ A∗ (X). Lemma 8.1. For every abelian variety X there exists a positive integer NX , such that for all x ∈ A1 (X), we have NX exp(logA (x)) ∈ A∗ (X) Proof. Let x ∈ A1 (X). Note that ∀m > g, xm = 0. Thus, by Lemma 2.8, logA (x) =

g−1 X

ηi xi+1 ∈

i=0

A1Q (X). We also get from Lemma 2.8 that βi = (i + 1)ηi ∈ A−i (X). Thus, logA (x)g logA (x)2 + ··· + 2! g! βi i+1 1 X βi βj x + xi+j+2 + i+1 2! (i + 1)(j + 1)

exp(logA (x)) =1 + logA (x) + =1 +

X 06i6g−1

+ ··· +

1 k!

i,j>0 i+j6g−2

X P ij >0 ij 6g−k

P βi1 · · · βik xg xk+ ij + · · · + . (i1 + 1) · · · (ik + 1) g!

Note that using the Lagrange multiplier rule, we may show that for any 1 ≤ k ≤ g, g x g k ≥ max (i1 + 1) · · · (ik + 1). Consider the function f (x) = defined on (0, g]. Using k x P ij >0 ij 6g−k

calculus, it is easy to check that this function attains a maximum at x = ge−1 . Thus, eg/e ≥ g k max implying that k k beg/e c ≥ max

max (i1 + 1) · · · (ik + 1).

16k6g P ij >0 ij 6g−k

Thus, g!beg/e c! is divisible by k!(i1 + 1) · · · (ik + 1) for all k and all sets {i1 , . . . , ik }. This means that we may take NX = g!beg/e c! so that NX exp(logA (x)) ∈ A∗ (X). ˆ as Definition 8.2. We define the integral Fourier-Mukai transformation FZA : A∗ (X) → A∗ (X) A A t ˆ (i) FZ := RA (NX×Xˆ CA (P)), (ii) FZ := RA (NX×Xˆ CA (P) ). Note that, this implies that FZA (x) = NX×Xˆ F A (x) and FˆZA (y) = NX×Xˆ F A (y). We get the following easy consequences of the properties of F A . Proposition 8.3. For any x, y ∈ A∗ (X), we have g ∗ 2 (2) NX×Xˆ FZA (x ? y) = FZA (x)FZA (y). (1) FˆZA ◦ FZA (x) = NX× ˆ (−1) σX (x), and X Proof. Using the fact that FZA = NX×Xˆ · F A , (1) follows from Theorem 4.3 and (2) follows from Proposition 5.2, part (1). 9. Consequences for algebraic cobordism Let X be an abelian variety over k of dimension g. Let I ⊂ CH g (X) denote the set of 0cycles of degree 0 on X. In [4, § 4], Bloch showed that I ?(r+1) ? CH r (X) = (0) in the cases r = 0, 1, g − 2, g − 1, g. In [2], Beauville conjectured that (Fp )

ˆ For all x ∈ CHQp (X), we have F(x) ∈ CHQ>g−p (X).

He verified (Fp ) for p = 0, 1, g − 2, g − 1, g ([2, Proposition 8.(i)]) and also showed that

FOURIER-MUKAI TRANSFORMATION ON ALGEBRAIC COBORDISM

15

Proposition 9.1 ([2, Proposition 9]). (Fp ) implies that I ?(p+1) ? CH p (X) = (0). In particular, the groups I ?(g+1) , I ?g ? CH g−1 (X) and I ?(g−1) ? CH g−2 (X) are zero. We prove an analogue of this Proposition replacing I with numerically trivial cobordism cycles. A notion of numerical equivalence on Ω∗ (X) was defined in [1]. We briefly recall the construction. Definition 9.2. Let Y be a smooth projective scheme over a field k of characteristic 0. Consider πY ∗ Ωm+n−dimY (k), where πY is the structhe composition of maps Ωm (Y ) ⊗ Ωn (Y ) → Ωm+n (Y ) −→ ture morphism Y → Spec k. This gives a map of L-modules Ω∗ (Y ) −→ HomL (Ω∗ (Y ), Ω∗ (k)). We say that a cobordism cycle in Ω∗ (Y ) is numerically equivalent to 0 if it is in N ∗ (Y ), which is the kernel of this map. We denote Ω∗num (Y ) := Ω∗ (Y )/N ∗ (Y ). Let A∗ (Y ) denote the subgroup of Ω∗ (Y ) consisting of cobordism cycles algebraically equivalent to zero. For the definition we refer the reader to [7, § 3]. It follows from [7, Theorem 10.3] and [1, Theorem 5.3] that A∗ (Y ) ⊆ N ∗ (Y ). Let us now go back to the special case of abelian varieties. ˆ Hence, F Ω maps N ∗ (X) to N ∗ (X). ˆ Lemma 9.3. FZΩ maps N ∗ (X) to N ∗ (X). Q Q ∗ Proof. Let α ∈ N ∗ (X) and N = NX×Xˆ . Then, by definition, FZΩ (α) = pX∗ ˆ (pX α · N CΩ (P)). ˆ and X respectively and let p ˆ and pX be the Let πXˆ and πX be the structure morphisms of X X ˆ ˆ ˆ we respective projections of X × X to X and X. From the projection formula for any γ ∈ Ω∗ (X) obtain Ω πX∗ p∗X α · N CΩ (P) · p∗Xˆ γ ˆ (FZ (α) · γ) = πX∗ ˆ pX∗ ˆ = πX∗ pX∗ p∗X α · N CΩ (P) · p∗Xˆ γ = πX∗ α · FˆZΩ (γ) .

Thus, numerical triviality of α implies that FZΩ (α) is numerically trivial as well.

Proposition 9.4. Fix 0 ≤ p ≤ g. If x ∈ ΩpQ (X) satisfies F Ω (x) ∈ Ω>g−p (X), then Q NQ∗ (X)?(p+1) ? x = 0. In particular, N ∗ (X)?(g+1) is torsion. Proof. Pick α1 , . . . , αp+1 ∈ N ∗ (X) and let N = NX×Xˆ . Note that by Proposition 8.3, part (2), (9.1)

N p FZΩ (α1 ? α2 ? · · · ? αp+1 ) = FZΩ (α1 )FZΩ (α2 ) · · · FZΩ (αp+1 ).

By the Generalized degree formula ([10, Theorem 4.4.7]), we get for any i, X ˆ FZΩ (αi ) = deg(FZΩ (αi ))[IdXˆ ] + ωZ [Z˜ → X], codimX ˆ Z>0

ˆ Z˜ is smooth with a birational morphism where the sum is over closed integral subschemes Z ⊂ X, ∗ Ω ˜ ˆ Then, by [1, Proposition 3.4], Z → Z and ωZ ∈ L . Lemma 9.3 shows that FZ (αi ) ∈ NQ∗ (X). ˆ deg(FZΩ (αi )) = 0. Hence, F Ω (αi ) ∈ L∗ · Ω>1 (X). p Ω ˆ For p = g, this implies that By (9.1), it follows that N FZ (α1 ? · · · ? αp+1 ) is in L∗ · Ω>p+1 (X). g Ω Ω N FZ (α1 ? · · · ? αg+1 ) = 0. Applying FˆZ , we get by Proposition 8.3, part (1), that N g+2 α1 ? · · · ? αg+1 = 0. Thus, N ∗ (X)?(g+1) is torsion. By Proposition 5.2, part (2), we also get N p F Ω (α1 ? · · · ? αp+1 ? x) = N p F Ω (α1 ? · · · ? αp+1 )F Ω (x) = 0. Applying Fˆ Ω and using Proposition 5.2, part (1), we get N p (−1)g (σX )∗ (α1 ? · · · ? αp+1 ? x) = 0. Hence, α1 ? · · · ? αp+1 ? x = 0 ∈ Ω∗Q (X), which completes the proof.

Corollary 9.5. A∗ (X)?(g+1) = 0. g+2 ∗ ?(g+1) Proof. Since A∗ (X) ⊆ N ∗ (X), we get by Proposition 9.4 that NX× = 0. But, [7, ˆ A (X) X

Theorem 8.4] implies that A∗ (X) is divisible. Thus, A∗ (X)?(g+1) = 0.

16

ANANDAM BANERJEE AND THOMAS HUDSON

Remark 9.6. One can check that N ∗ (X) forms an ideal of Ω∗ (X) under Pontryagin product. By ∼ [10, Lemma 4.5.10] and [1, Theorem 3.2], N g (X) → I, which is the subgroup of 0-cycles of degree 0. g g ?2 ∼ ?2 ∼ This implies N (X)/N (X) → I/I → X. It would be interesting to study the structure of the groups N p (X)/N p (X)?2 for p < g. Acknowledgments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 2013-042157). We would like to thank Jinhyun Park for the several useful discussions and his kind support. References [1] A. Banerjee and J. Park, On numerical equivalence for algebraic cobordism, preprint available at http:// arxiv.org/abs/1312.1787, 2013. [2] A. Beauville, Quelques remarques sur la transformation de Fourier dans lanneau de Chow dune vari´ et´ e ab´ elienne, Algebraic geometry, LNM 1016 (1983), 238–260. [3] , Sur l’anneau de Chow d’une vari´ et´ e ab´ elienne, Mathematische Annalen 273 (1986), 647–651. [4] S. Bloch, Some Elementary Theorems about Algebraic Cycles on Abelian Varieties, Inventiones math. 37 (1976), 215–228. [5] C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201–219. [6] W. Fulton, Intersection theory, second edition ed., Ergebnisse der Math. Grenzgebiete, vol. 3, Springer-Verlag, Berlin, 1998. [7] A. Krishna and J. Park, Algebraic cobordism theory attached to algebraic equivalence, J. K-theory 11 (2013), 73–112. [8] P. S. Landweber, K¨ unneth formulas for bordism theories, Transactions of the American Mathematical Society (1966), 242–256. [9] M. Lazard, Sur les groupes de Lie formels a ´ un param` etre, Bull. Soc. Math. France 83 (1955), 251–274. MR 0073925 (17,508e) [10] M. Levine and F. Morel, Algebraic Cobordism, Springer Monographs Math., Berlin, 2007. [11] Ju. I. Manin, Correspondences, motifs and monoidal transformations, Math. USSR-Sb. 6 (1968), 439–470. ˆ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), [12] S. Mukai, Duality between D(X) and D(X) 153–175. [13] D. Mumford, C. P. Ramanujam, and Y. I. Manin, Abelian varieties, vol. 108, Oxford Univ Press, 1974. [14] A. Nenashev and K. Zainoulline, Oriented cohomology and motivic decompositions of relative cellular spaces, J. Pure Appl. Algebra 205 (2006), 323–340. [15] D. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56. MR 0290382 (44 #7566) [16] A. J. Scholl, Classical motives, Motives, Proc. Sympos. Pure Math. 55 (1994), 163–187. Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea (South) E-mail address: [email protected] URL: http://mathsci.kaist.ac.kr/~anandam/ Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea (South) E-mail address: [email protected]